General convergence conditions of Newton's method for m-Fréchet differentiable operators

Resumen

We present a local as well as a semilocal convergence analysis for Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149-159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391-400, 2009, Argyros and Hilout in J. Complex. 28:364-387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131-145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612-621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250-258, 1979. Numerical examples are presented further validating the theoretical results. © 2013 Korean Society for Computational and Applied Mathematics.